Problem: Simplify the following expression and state the condition under which the simplification is valid. $r = \dfrac{3t^2 + 45t + 150}{-6t^3 - 24t^2 + 30t}$
First factor out the greatest common factors in the numerator and in the denominator. $ r = \dfrac {3(t^2 + 15t + 50)} {-6t(t^2 + 4t - 5)} $ $ r = -\dfrac{3}{6t} \cdot \dfrac{t^2 + 15t + 50}{t^2 + 4t - 5} $ Simplify: $ r = - \dfrac{1}{2t} \cdot \dfrac{t^2 + 15t + 50}{t^2 + 4t - 5}$ Next factor the numerator and denominator. $ r = - \dfrac{1}{2t} \cdot \dfrac{(t + 5)(t + 10)}{(t + 5)(t - 1)}$ Assuming $t \neq -5$ , we can cancel the $t + 5$ $ r = - \dfrac{1}{2t} \cdot \dfrac{t + 10}{t - 1}$ Therefore: $ r = \dfrac{ -t - 10 }{ 2t(t - 1)}$, $t \neq -5$